overview two paired samples: within-subject designs -hypothesis test -confidence interval -effect...

Overview

• Two paired samples: Within-Subject Designs-Hypothesis test

-Confidence Interval

-Effect Size

• Two independent samples: Between-Subject Designs-Hypothesis test

-Confidence interval

-Effect Size

Comparing Two Populations

Until this point, all the inferential statistics we have considered involve using one sample as the basis for drawing conclusion about one population.

Although these single sample techniques are used occasionally in real research, most research studies aim to compare of two (or more) sets of data in order to make inferences about the differences between two (or more) populations.

What do we do when our research question concerns a mean difference between two sets of data?

Two kinds of studies

There are two general research strategies that can be used to obtain the two sets of data to be compared:

1. The two sets of data could come from two independent populations (e.g. women and men, or students from section A and from section B)

2. The two sets of data could come from related populations (e.g. “before treatment” and “after treatment”)

<- between-subjects design

<- within-subjects design

Part I

• Two paired samples: Within-Subject Designs-Hypothesis test


-Effect Size

Paired T-Test for Within-Subjects Designs

Our hypotheses:

Ho: D = 0

HA: D 0

To test the null hypothesis, we’ll again compute a t statistic and look it up in the t table.

Paired Samples t

t = D - D sD = sD

s

n

Steps for Calculating a Test Statistic

Paired Samples T

1. Calculate difference scores

2. Calculate D

3. Calculate sd

4. Calculate T and d.f.

5. Use Table E.6

Confidence Intervals for Paired Samples

Paired Samples t

D t (sD)

General formula

X t (SE)

Effect Size for Dependent Samples

Paired Samples d

One Sample d

s

Xd H0ˆ

Ds

Dd ˆ

Exercise

In Everitt’s study (1994), 17 girls being treated for anorexia were weighed before and after treatment. Difference scores were calculated for each participant.

Change in Weight

n = 17 = 7.26sD = 7.16D

Test the null hypothesis that there was no change in weight.

Compute a 95% confidence interval for the mean difference.

Calculate the effect size

ExerciseChange in Weight

n = 17 = 7.26sD = 7.16D

74.117

16.7SE

17.474.1

026.7)16(

t

01.p

T-test


n = 17 = 7.26sD = 7.16D

12.2critt

)74.1(12.226.7 CI

)95.10,57.3(

Confidence Interval


n = 17 = 7.26sD = 7.16D

16.7

26.7d

01.1d

Effect Size

Part II

• Two independent samples: Between-Subject Designs-Hypothesis test


-Effect Size

T-Test for Independent SamplesThe goal of a between-subjects research study is to evaluate the mean difference between two populations (or between two treatment conditions).

Ho: 1 = 2

HA: 1 2

We can’t compute difference scores, so …

T-Test for Independent Samples

We can re-write these hypotheses as follows:

Ho: 1 - 2 = 0

HA: 1 - 2 0

To test the null hypothesis, we’ll again compute a t statistic and look it up in the t table.


General t formula

t = sample statistic - hypothesized population parameter estimated standard error

?21 XXs

Independent samples t

21

)()( 2121

XXs

XXt

One Sample t

X

Htest s

Xt 0


Standard Error for a Difference in Means

The single-sample standard error ( sx ) measures how much error expected between X and .

The independent-samples standard error (sx1-x2) measures how much error is expected when you are using a sample mean difference (X1 – X2) to represent a population mean difference.

21 XXs



Each of the two sample means represents its own population mean, but in each case there is some error.

The amount of error associated with each sample mean can be measured by computing the standard errors.

To calculate the total amount of error involved in using two sample means to approximate two population means, we will find the error from each sample separately and then add the two errors together.

2

22

1

21

21 n

s

n

ss XX



But…

This formula only works when n1 = n2. When the two samples are different sizes, this formula is biased.

This comes from the fact that the formula above treats the two sample variances equally. But we know that the statistics obtained from large samples are better estimates, so we need to give larger sample more weight in our estimated standard error.

2

22

1

21

21 n

s

n

ss XX



We are going to change the formula slightly so that we use the pooled sample variance instead of the individual sample variances.

This pooled variance is going to be a weighted estimate of the variance derived from the two samples.

sp2

SS1 SS2

df1 df2

2

2

1

2

21 n

s

n

ss pp

XX


One-Sample T

1. Calculate sample mean

2. Calculate standard error


4. Use Table D

Independent Samples T

1. Calculate X1-X2

2. Calculate pooled variance

3. Calculate standard error


5. Use Table E.6

sp2

SS1 SS2

df1 df2

sp2

n1

sp

2

n2

t (X 1 X 2) (1 2 )

sx 1 x 2 d.f. = (n1 - 1) + (n2 - 1)


Illustration

A developmental psychologist would like to examine the difference in verbal skills for 8-year-old boys versus 8-year-old girls. A sample of 10 boys and 10 girls is obtained, and each child is given a standardized verbal abilities test. The data for this experiment are as follows:

Girls Boys

n1 = 10 = 37SS1 = 150

X 1

n2 = 10 = 31SS2 = 210

X 2

STEP 1: get mean difference

621 XX

Girls Boys

n1 = 10 = 37SS1 = 150

X 1

n2 = 10 = 31SS2 = 210

X 2

Illustration

STEP 2: Compute Pooled Variance

sp2

SS1 SS2

df1 df2

150 210

(10 1) (10 1)

360

1820

Girls Boys

n1 = 10 = 37SS1 = 150

X 1

n2 = 10 = 31SS2 = 210

X 2

Illustration

STEP 3: Compute Standard Error

Girls Boys

n1 = 10 = 37SS1 = 150

X 1

n2 = 10 = 31SS2 = 210

X 2

SE s p

2

n1

s p

2

n2

20

10

20

10 4 2

Illustration

STEP 4: Compute T statistic and df

Girls Boys

n1 = 10 = 37SS1 = 150

X 1

n2 = 10 = 31SS2 = 210

X 2

t (X 1 X 2) (1 2 )

sx 1 x 2

(37 31) 0

23

d.f. = (n1 - 1) + (n2 - 1) = (10-1) + (10-1) = 18

Illustration

STEP 5: Use table E.6

Girls Boys

n1 = 10 = 37SS1 = 150

X 1

n2 = 10 = 31SS2 = 210

X 2

T = 3 with 18 degrees of freedom

For alpha = .01, critical value of t is 2.878

Our T is more extreme, so we reject the null

There is a significant difference between boys and girls

Illustration


Sample Data

Hypothesized Population Parameter

Sample Variance

Estimated Standard

Error

t-statistic

Single sample

t-statistic

Independent samples t-statistic

X 1 X 2

1 2

sp2

n1

sp

2

n2

sp2

SS1 SS2

df1 df2

X

s2

n

s2 SS

df

t X

sx

t (X 1 X 2) (1 2 )

sx 1 x 2

Confidence Intervals for Independent Samples

One Sample t

X t (sx)

General formula

X t (SE)

Independent Sample t

(X1-X2) t (sx1-x2)

Effect Size for Independent Samples

One Sample d

Independent Samples d

s

Xd H0ˆ

ps

XXd 21ˆ

Exercise

Subjects are asked to memorize 40 noun pairs. Ten subjects are given a heuristic to help them memorize the list, the remaining ten subjects serve as the control and are given no help. The ten experimental subjects have a X-bar = 21 and a SS = 100. The ten control subjects have a X-bar = 19 and a SS = 120.

Test the hypothesis that the experimental group differs from the control group.

Give a 95% confidence interval for the difference between groups

Give the effect size

ExerciseExperimental Control

n1 = 10 = 21SS1 = 100

X 1

n2 = 10 = 19SS2 = 120

X 2

221 XX

2.1218

220

)110()110(

120100

21

212

dfdf

SSSSsp

56.144.210

2.12

10

2.12

2

2

1

2

n

s

n

sSE pp

T-test

Exercise

28.156.1

02)()(

21

2121

xxs

XXt

d.f. = (n1 - 1) + (n2 - 1) = (10-1) + (10-1) = 18

20.p

T-test

Experimental Control

n1 = 10 = 21SS1 = 100

X 1

n2 = 10 = 19SS2 = 120

X 2

Exercise

101.2critt

)56.1(101.22 CI

Confidence Interval

)28.5,28.1(


n1 = 10 = 21SS1 = 100

X 1

n2 = 10 = 19SS2 = 120

X 2

Exercise

57.d

Effect Size

ps

XXd 21

2.12

2d


n1 = 10 = 21SS1 = 100

X 1

n2 = 10 = 19SS2 = 120

X 2

Summary

Hypothesis Tests

Confidence Intervals

Effect Sizes

1 Sample

2 Paired Samples

2 Independent Samples

Review

Sample Data

Hypothesized Population Parameter

Sample Variance

Estimated Standard

Error

t-statistic

One sample

t-statistic

Paired samples t-

statistic

Independent samples t-statistic

X

X 1 X 2

1 2

s2

n

sp2

n1

sp

2

n2

s2 SS

df

sp2

SS1 SS2

df1 df2

t X

sx

t (X 1 X 2) (1 2 )

sx 1 x 2

D D

s2

ndf

SSs D2

D

D

s

Dt

Confidence Intervals

Paired Samples t

D t (sD)

One Sample t

X t (SE)

Independent Sample t

(X1-X2) t (sx1-x2)

Effect Sizes

Paired Samples d

One Sample d

Independent Samples d

Ds

Dd ˆ

s

Xd H0ˆ

ps

XXd 21ˆ

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